Exponentials & Logarithms

Inverse growth languages

Exponential y = aˣ grows (or decays) by a constant factor. Logarithm logₐ(x) answers: to what power must a be raised to get x?

Solving patterns

• Same base: aᶠ⁽ˣ⁾ = aᵍ⁽ˣ⁾ ⇒ f(x) = g(x).
• Take ln both sides when bases differ.
• Exponential growth: A = A₀ eᵏᵗ or A₀(1+r)ᵗ.
• Half-life: A = A₀ (1/2)^(t/T).

Worked example

Solve 2ˣ = 10. Take ln: x ln 2 = ln 10 → x = ln 10 / ln 2 ≈ 3.32.